with the \(\bf {\varepsilon }_{i,h}\) 's being independent centered Gaussian vectors of errors with covariance \(\sigma ^2I_D\) . The errors are assumed to be independent of the \(\textbf {Z}_i\) 's and the \(\textbf {Z}_i\) 's are independent and identically distributed with the same distribution as \(\textbf {Z}\) . Here, we assume that the grid is fixed and regular with \(p\)  points, note that our approach could be generalized to the case where \(|t_h-h/p|\le 1/p\) . With the modern ability to record multiple data streams at a very fine timescale, the need for new tools to perform analysis on all these streams at once has arisen. From a functional data analysis (FDA) perspective, these data can be considered as realizations of a multivariate random process on a dense grid. In this context, Multivariate functional principal components analysis (MfPCA) provides an interesting analysis framework.
As MfPCA is an extension of functional principal component analysis (fPCA) to the multivariate case, it inherits most of its properties
regarding representation. Thus as fPCA, it is used as a preprocessing tool. For example, we can cite the modeling and forecasting of multi-population mortality (see [1]}) and identification of biomarkers for accurate diagnosis of Alzheimer's disease in an early stage (see [2]}). The notable advantage of such an approach is its capacity to handle all the data presented at once, i.e., accounting for variability through time (intra-functional correlations) and space (correlations between functions).
